Question A.2: Determine x1 , x2 , and x3 for this set of simultaneous equa......
Determine x_{1} , x_{2} , and x_{3} for this set of simultaneous equations:
25x_{1} – 5x_{2} – 20x_{3} = 50
-5x_{1} + 10x_{2} – 4x_{3} = 0
– 5x_{1} – 4x_{2} + 9x_{3} = 0
In matrix form, the given set of equations becomes
\begin{bmatrix} 25 & -5 & -20 \\ -5 & 10 & -4 \\ -5 & -4 & 9 \end{bmatrix} \begin{bmatrix} x_{1} \\ x_{2}\\ x_{3} \end{bmatrix} = \begin{bmatrix} 50 \\ 0\\ 0 \end{bmatrix}
We apply Eq. (A.11) to find the determinants. This requires that we repeat the first two rows of the matrix. Thus,
= a_{11} a_{22} a_{33} + a_{21} a_{32} a_{13} + a_{31} a_{12} a_{23} – a_{13} a_{22} a_{31} – a_{23} a_{32} a_{11} – a_{33} a_{12} a_{21} (A.11)
Δ = \begin{vmatrix} 25 & -5 & -20 \\ -5 & 10 & -4 \\ -5 & -4 & 9 \end{vmatrix} =
= 25(10)9 + (- 5) (- 4)(- 20) + (- 5)(- 5)(- 4)
– (-20)(10)(-5) – (- 4)(- 4)25 – 9(- 5)(- 5)
= 2250 – 400 – 100 – 1000 – 400 – 225 = 125
Similarly,
Δ_{1} = \begin{vmatrix} 50 & -5 & -20 \\ 0 & 10 & -4 \\ 0 & -4 & 9 \end{vmatrix} = 
= 4500 + 0 + 0 – 0 – 800 – 0 = 3700
Δ_{2} = \begin{vmatrix} 25 & 50 & -20 \\ -5 & 0 & -4 \\ -5 & 0 & 9 \end{vmatrix} =
= 0 + 0 + 1000 – 0 – 0 + 2250 = 3250
Δ_{3} = \begin{vmatrix} 25 & -5 & 50 \\ -5 & 10 & 0 \\ -5 & -4 & 0 \end{vmatrix} =
= 0 + 1000 + 0 + 2500 – 0 – 0 = 3500
Hence, we now find
x_{1} = \frac{Δ_{1}}{Δ} = \frac{3700}{125} = 29.6
x_{2} = \frac{Δ_{2}}{Δ} = \frac{3250}{125} = 26
x_{3} = \frac{Δ_{2}}{Δ} = \frac{3500}{125} = 28